Skew prime polynomial matrices

A pair of polynomial matrices, and , is defined to be "externally skew prime" if and only if a solution, , to the polynomial matrix equation exists. It is shown that and are externally skew prime if and only if with and relatively left prime and and relatively right prime. This observation implies a new constructive procedure for determining and where and are found to be externally skew prime and is nonsingular. A new procedure for obtaining solutions to the more general polynomial matrix equation, , based on the notion of skew-prime polynomial matrices is also presented. A characterization of all solutions when is also given, under appropriate assumptions, and then employed to determine a unique solution to this polynomial matrix equation.